∫ (1−2x)3(2+3x)6 ________________________

3.13.64 \(\int \frac {(1-2 x)^3 (2+3 x)^6}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=76 \[ -\frac {729 x^8}{25}-\frac {37908 x^7}{875}+\frac {12231 x^6}{625}+\frac {774981 x^5}{15625}-\frac {5643 x^4}{3125}-\frac {1836723 x^3}{78125}-\frac {461623 x^2}{390625}+\frac {13880997 x}{1953125}-\frac {1331}{9765625 (5 x+3)}+\frac {23232 \log (5 x+3)}{9765625} \]

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Rubi [A] time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {88} \begin {gather*} -\frac {729 x^8}{25}-\frac {37908 x^7}{875}+\frac {12231 x^6}{625}+\frac {774981 x^5}{15625}-\frac {5643 x^4}{3125}-\frac {1836723 x^3}{78125}-\frac {461623 x^2}{390625}+\frac {13880997 x}{1953125}-\frac {1331}{9765625 (5 x+3)}+\frac {23232 \log (5 x+3)}{9765625} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[((1 - 2*x)^3*(2 + 3*x)^6)/(3 + 5*x)^2,x]

[Out]

(13880997*x)/1953125 - (461623*x^2)/390625 - (1836723*x^3)/78125 - (5643*x^4)/3125 + (774981*x^5)/15625 + (122

31*x^6)/625 - (37908*x^7)/875 - (729*x^8)/25 - 1331/(9765625*(3 + 5*x)) + (23232*Log[3 + 5*x])/9765625

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI

ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&

(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {(1-2 x)^3 (2+3 x)^6}{(3+5 x)^2} \, dx &=\int \left (\frac {13880997}{1953125}-\frac {923246 x}{390625}-\frac {5510169 x^2}{78125}-\frac {22572 x^3}{3125}+\frac {774981 x^4}{3125}+\frac {73386 x^5}{625}-\frac {37908 x^6}{125}-\frac {5832 x^7}{25}+\frac {1331}{1953125 (3+5 x)^2}+\frac {23232}{1953125 (3+5 x)}\right ) \, dx\\ &=\frac {13880997 x}{1953125}-\frac {461623 x^2}{390625}-\frac {1836723 x^3}{78125}-\frac {5643 x^4}{3125}+\frac {774981 x^5}{15625}+\frac {12231 x^6}{625}-\frac {37908 x^7}{875}-\frac {729 x^8}{25}-\frac {1331}{9765625 (3+5 x)}+\frac {23232 \log (3+5 x)}{9765625}\\ \end {align*}

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Mathematica [A] time = 0.05, size = 71, normalized size = 0.93 \begin {gather*} \frac {-49833984375 x^9-103939453125 x^8-10979296875 x^7+104830031250 x^6+47772112500 x^5-42029925000 x^4-26126590000 x^3+10934112000 x^2+10818777780 x+813120 (5 x+3) \log (6 (5 x+3))+2118706028}{341796875 (5 x+3)} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[((1 - 2*x)^3*(2 + 3*x)^6)/(3 + 5*x)^2,x]

[Out]

(2118706028 + 10818777780*x + 10934112000*x^2 - 26126590000*x^3 - 42029925000*x^4 + 47772112500*x^5 + 10483003

1250*x^6 - 10979296875*x^7 - 103939453125*x^8 - 49833984375*x^9 + 813120*(3 + 5*x)*Log[6*(3 + 5*x)])/(34179687

5*(3 + 5*x))

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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^3 (2+3 x)^6}{(3+5 x)^2} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

IntegrateAlgebraic[((1 - 2*x)^3*(2 + 3*x)^6)/(3 + 5*x)^2,x]

[Out]

IntegrateAlgebraic[((1 - 2*x)^3*(2 + 3*x)^6)/(3 + 5*x)^2, x]

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fricas [A] time = 1.05, size = 67, normalized size = 0.88 \begin {gather*} -\frac {9966796875 \, x^{9} + 20787890625 \, x^{8} + 2195859375 \, x^{7} - 20966006250 \, x^{6} - 9554422500 \, x^{5} + 8405985000 \, x^{4} + 5225318000 \, x^{3} - 2186822400 \, x^{2} - 162624 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 1457504685 \, x + 9317}{68359375 \, {\left (5 \, x + 3\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^6/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/68359375*(9966796875*x^9 + 20787890625*x^8 + 2195859375*x^7 - 20966006250*x^6 - 9554422500*x^5 + 8405985000

*x^4 + 5225318000*x^3 - 2186822400*x^2 - 162624*(5*x + 3)*log(5*x + 3) - 1457504685*x + 9317)/(5*x + 3)

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giac [A] time = 0.93, size = 102, normalized size = 1.34 \begin {gather*} \frac {1}{341796875} \, {\left (5 \, x + 3\right )}^{8} {\left (\frac {422820}{5 \, x + 3} - \frac {2021355}{{\left (5 \, x + 3\right )}^{2}} + \frac {474957}{{\left (5 \, x + 3\right )}^{3}} + \frac {9876195}{{\left (5 \, x + 3\right )}^{4}} + \frac {14499345}{{\left (5 \, x + 3\right )}^{5}} + \frac {10904215}{{\left (5 \, x + 3\right )}^{6}} + \frac {5836215}{{\left (5 \, x + 3\right )}^{7}} - 25515\right )} - \frac {1331}{9765625 \, {\left (5 \, x + 3\right )}} - \frac {23232}{9765625} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^6/(3+5*x)^2,x, algorithm="giac")

[Out]

1/341796875*(5*x + 3)^8*(422820/(5*x + 3) - 2021355/(5*x + 3)^2 + 474957/(5*x + 3)^3 + 9876195/(5*x + 3)^4 + 1

4499345/(5*x + 3)^5 + 10904215/(5*x + 3)^6 + 5836215/(5*x + 3)^7 - 25515) - 1331/9765625/(5*x + 3) - 23232/976

5625*log(1/5*abs(5*x + 3)/(5*x + 3)^2)

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maple [A] time = 0.01, size = 57, normalized size = 0.75 \begin {gather*} -\frac {729 x^{8}}{25}-\frac {37908 x^{7}}{875}+\frac {12231 x^{6}}{625}+\frac {774981 x^{5}}{15625}-\frac {5643 x^{4}}{3125}-\frac {1836723 x^{3}}{78125}-\frac {461623 x^{2}}{390625}+\frac {13880997 x}{1953125}+\frac {23232 \ln \left (5 x +3\right )}{9765625}-\frac {1331}{9765625 \left (5 x +3\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3*(3*x+2)^6/(5*x+3)^2,x)

[Out]

13880997/1953125*x-461623/390625*x^2-1836723/78125*x^3-5643/3125*x^4+774981/15625*x^5+12231/625*x^6-37908/875*

x^7-729/25*x^8-1331/9765625/(5*x+3)+23232/9765625*ln(5*x+3)

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maxima [A] time = 0.53, size = 56, normalized size = 0.74 \begin {gather*} -\frac {729}{25} \, x^{8} - \frac {37908}{875} \, x^{7} + \frac {12231}{625} \, x^{6} + \frac {774981}{15625} \, x^{5} - \frac {5643}{3125} \, x^{4} - \frac {1836723}{78125} \, x^{3} - \frac {461623}{390625} \, x^{2} + \frac {13880997}{1953125} \, x - \frac {1331}{9765625 \, {\left (5 \, x + 3\right )}} + \frac {23232}{9765625} \, \log \left (5 \, x + 3\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^6/(3+5*x)^2,x, algorithm="maxima")

[Out]

-729/25*x^8 - 37908/875*x^7 + 12231/625*x^6 + 774981/15625*x^5 - 5643/3125*x^4 - 1836723/78125*x^3 - 461623/39

0625*x^2 + 13880997/1953125*x - 1331/9765625/(5*x + 3) + 23232/9765625*log(5*x + 3)

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mupad [B] time = 0.05, size = 54, normalized size = 0.71 \begin {gather*} \frac {13880997\,x}{1953125}+\frac {23232\,\ln \left (x+\frac {3}{5}\right )}{9765625}-\frac {1331}{48828125\,\left (x+\frac {3}{5}\right )}-\frac {461623\,x^2}{390625}-\frac {1836723\,x^3}{78125}-\frac {5643\,x^4}{3125}+\frac {774981\,x^5}{15625}+\frac {12231\,x^6}{625}-\frac {37908\,x^7}{875}-\frac {729\,x^8}{25} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x - 1)^3*(3*x + 2)^6)/(5*x + 3)^2,x)

[Out]

(13880997*x)/1953125 + (23232*log(x + 3/5))/9765625 - 1331/(48828125*(x + 3/5)) - (461623*x^2)/390625 - (18367

23*x^3)/78125 - (5643*x^4)/3125 + (774981*x^5)/15625 + (12231*x^6)/625 - (37908*x^7)/875 - (729*x^8)/25

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sympy [A] time = 0.13, size = 68, normalized size = 0.89 \begin {gather*} - \frac {729 x^{8}}{25} - \frac {37908 x^{7}}{875} + \frac {12231 x^{6}}{625} + \frac {774981 x^{5}}{15625} - \frac {5643 x^{4}}{3125} - \frac {1836723 x^{3}}{78125} - \frac {461623 x^{2}}{390625} + \frac {13880997 x}{1953125} + \frac {23232 \log {\left (5 x + 3 \right )}}{9765625} - \frac {1331}{48828125 x + 29296875} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**3*(2+3*x)**6/(3+5*x)**2,x)

[Out]

-729*x**8/25 - 37908*x**7/875 + 12231*x**6/625 + 774981*x**5/15625 - 5643*x**4/3125 - 1836723*x**3/78125 - 461

623*x**2/390625 + 13880997*x/1953125 + 23232*log(5*x + 3)/9765625 - 1331/(48828125*x + 29296875)

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